binary relations defining an equivalence relation on S

Question

Is this a true statement for binary relations defines an equivalence relation on S:

S is the set of all n-digit binary sequences. We say that two binary sequences are in a relation if and only if they have the same number of 1s.

Answer 1

If $f:S\to X$ is some function then the relation $\sim$ on $S$ prescribed by: $$s\sim s'\iff f(s)=f(s')$$ is an equivalence relation. This simply because for all $s,t,r\in S$:

• $f(s)=f(s)$ (reflexivity)
• $f(s)=f(t)\implies f(t)=f(s)$ (symmetry)
• $f(s)=f(t)\wedge f(t)=f(r)\implies f(s)=f(r)$ (transitivity)

In your question let $f$ be the function that sends each sequence to its number of $1$'s.